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AMORTIZATIO  N 


A  Guide  to  the  Ready  Computation  of  the 

Investment  Value  of  Bonds  by  the  use 

of  the  Extended  Bond  Tables 


By  Charles  E.  Sprague 


second  edition 


New  York,  1910 
pubusht  by  the  author 


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GEOIGE  H/RJES  CO. 

35  V^EST  31si  STREET 

NEW  YORK,  N.  Y. 


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AMORTIZATION 


A  GuiDK  TO  THE  Ready  Computation  of  the 

Investment  Value  of  Bonds  by  the  use 

OF  THE  Extended  Bond  Tables 


By  Charles  E.  Spragub 


second  edition 


New  York,  1910 
publisht  by  the  author 


< 


Copyright,  1908,  by  Charles  E.  Sprague 


PREFACE. 


TDELIEVING  that  the  principle  of  amortization  was  the  only  true 
method  of  valuing  securities  held  for  investment,  and  that  ulti- 
mately it  would  be  the  one  ofiScially  adopted,  I  commenced  in  1898  to 
make  the  necessary  computations  for  placing  the  securities  of  this  savings 
bank  on  that  basis.  I  found  that  the  existing  tables  of  bond  values  were 
useless  for  that  purpose,  as  their  decimals  are  not  suflBciently  extended, 
giving  accurate  values  to  $100  only.  I  succeeded  at  the  beginning  of 
1901  in  inaugurating  the  system  of  amortization,  after  an  enormous 
amount  of  labor,  but  with  conscientious  accuracy.  Realizing  the  need 
for  extended  tables  which  would  greatly  facilitate  the  work,  and  utilizing 
the  experience  and  some  of  the  results  already  obtained,  I  then  took  up 
the  still  greater  task  of  making  all  the  calculations  for  a  set  of  tables 
which  would  have  saved  me  nine-tenths  of  my  labor  had  they  been  in 
existence  in  1898.  These  tables  were  issued  in  1905,  and  are  extended  to 
eight  places  of  decimals. 

As  amortization  has  now  been  enacted  into  law,  I  have  prepared 
this  little  treatise  as  a  companion  to  the  Tables,  hoping  it  may  be  useful 
to  all  officers  of  financial  institutions  and  may  aid  them  in  their  labors. 
I  have  explained  the  theory  in  my  book  entitled  "Text-Book  of  the 
Accountancy  of  Investment,  ' '  but  even  those  who  have  studied  the 
theory  are  benefited  by  having  at  hand  a  simple  set  of  rules. 

I  am  satisfied  that  by  the  use  of  the  Extended  Tables,  guided  by 
these  rules,  scientific  amortization,  within  the  conditions  laid  down  by 
the  Banking  Department,  will  be  much  easier  of  operation  than  the  so- 
called  •*  simple  "  but  very  inaccurate  method  prescribed  as  an  altprnativ. 


CHARIvES  E.  SPRAGUE. 


Union  Dime  Savings  Bank, 
New  York,  June,  1908. 


247038 


CONTENTS. 


PAGE 

Introductory 1 

What  is  Investment  Vai.ue  ? 1 

Prewminary  Steps 3 

Determining  the  Basis 3 

1.  For  Even  Half- Years 4 

2.  For  Broken  Times 4 

3.  For  Serial  or  Various  Maturities C 

4.  Bonds  at  Unusual  Rates 6 

The  Inaugurai,  Vai^ue 7 

1.  For  J.  &  J.  Bonds 7 

2.  For  Bonds  at  other  Dates 7 

3.  For  Serial  Bonds 8 

4.  Combining  Several  Purchases.. 10 

Continuing  the  Amortization 10 

1.  For  J.  &  J.  Bonds 11 

Schedule  of  Amortization 12 

2.  For  Bonds  at  other  Dates 13 

3.  Serial  Bonds 18 

New  Purchases 19 

Intermediate  Rates 19 

OuARTERi,Y  Bonds 21 

Annuai,  Bonds 21 

Residues 21 

Accrued  Interest 22 

The  Proper  Basis  of  Bond  Accounts  vi^hen  held  for  Invest- 
ment    23 


AMORTIZATION 


The  Banking  Law  of  the  State  of  New  York  as  amended,  paragraph 
20,  makes  it  the  duty  of  each  Savings  Bank  to  report  semi-annually  to 
the  Superintendent  of  Banks,  among  other  things,  "  the  estimated  invest- 
ment value  of  all  stock  and  bond  investments"  "in  the  manner  pre- 
scribed by  the  Superintendent  of  Banks." 

In  accordance  with  this  provision,  the  Superintendent  has  Qune, 
1908)  issued  a  circular  in  which  he  recognizes  as  admissible  two  methods 
of  ascertaining  investment  values,  designated  as  the  scientific  and  the 
simple,  or  pro  rata,  method,  respectivly. 

If  we  were  compelled  to  compute  each  separate  value  by  arithmetic, 
the  pro  rata  method  would  in  most  cases  be  easier;  but  where  the  values 
at  all  ordinary  rates  and  times  have  been  exactly  calculated  and  tabu- 
lated, as  they  are  in  Sprague's  Bond  Vai.uks,  it  is  quite  as  easy  to  use 
the  "scientific"  method. 

The  scientific  method  gives  far  more  satisfactory  results  and 
equalizes  the  rate  of  interest  during  the  life  of  the  bond.  If  the  "simple  " 
method  be  applied  to  a  seven-year  7%  bond  bought  on  a  4%  basis,  it  will 
be  found  that  the  income  on  the  investment  value,  instead  of  being  a 
uniform  4%,  varies  from  3.73%  to  4.36%.  It  does  not^  therefore,  "result 
in  giving  the  purchaser  the  exact  periodical  income  upon  the  basis  of 
which  the  bond  was  purchased,"  even  for  this  short  maturity. 

This  treatise  is  intended  to  furnish  rules  of  procedure  by  which  the 
scientific  investment-value  may  most  readily  be  ascertained  from  tables 
of  extended  bond-values;  the  "simple"  or  "pro  rata"  values  will  be 
disregarded 

While  very  little  of  the  theory  of  the  subject  need  be  introduced 
here,  a  definition  of  the  terms  used  will  be  proper. 


What  is  Investment  Value? 

The  income-basis  being  that  rate  of  compound  interest  which,  ap- 
plied to  the  original  cost,  will  produce  at  the  proper  times  all  the  pay- 
ments provided  for  in  the  security,  whether  designated  as  principal  or  as 
interest:  the  investment  value  is  that  value  which,  upon  the  original 
income  basis  at  the  time  of  investment,  will  amortize  or  gradually 
extinguish  the  premium  or  discount  so  as  to  bring  the  security  to  par  at 
maturity. 

The  successiv  values  of  bonds  under  these  definitions  are  obtain- 
able in  two  ways  : 

1.  By  discounting  to  the  present  date  at  true  compound  interest, 
on  the  income  basis,  every  future  payment  of  principal  or  interest. 


2  Amortization 

2.  By  applying  to  the  value  reported  at  the  next  previous  report 
the  amount  known  as  the  amortization. 

The  amortization  is  the  difference  between  the  nominal  and  the 
effectiv  interest  or  net  income. 

The  nominal  interest  is  a  constant  quantity  being  a  fixt  percent- 
age of  the  par  value. 

The  effectiv  interest  is  a  fixt  percentage  of  the  investment  value, 
but  as  this  constantly  varies,  approaching  to  par,  the  effective  interest 
also  varies  and  the  amortization,  whether  of  premiums  or  discounts, 
constantly  increases. 

•  Thus,  if  a  6%  bond  for  $10,000.,  interest  semi-annual,  was  pur- 
chased at  some  time  in  the  past  on  a  4%  basis  and  now  has  one  year  to 
run  to  maturity,  the  nominal  interest  is  |300.  each  half  year  ;  its  effectiv 
interest  is  2%  each  half  year  on  whatever  the  investment  value  may  be 
at  the  time.     The  income  basis  is  4%. 

The  investment  value  is  composed  of  the  following  values  : 

1.  Present  worth  of  $      300.  at  4%  >^  year  from  now . .    $      294.12 

2.  Present  worth  of         300.  at    ••  (cpd.)    1  year  from  now . .  288.35 

3.  Present  worth  of     10,000.  at    "       "        1  year  from  now . .       9,611.69 

ToTAi, $10,194.16 

This  result  is  corroborated  by  various  tables  of  bond  values. 

At  six  months  before  maturity  the  corresponding  figures  would  be: 

1.  Present  worth  of  §      300.  at  6  months $      294.12 

2.  Present  worth  of    10,C00.  at  6  months 9,803.92 

New  Investment  Vai,ue...  $10,098.04 


To  proceed  by  amortization,  we  begin  with  the  value  at  one 

year $10,194.16 

The  interest  for  6  months  on  this  investment  is  $203.88 

The  nominal  interest  is  3%  of  $10,000 $300.00 

The  effectiv  interest  is  2%  of  $10,194.16 203.88 

The  amortization  is  the  difference $  96.12 

Subtracting  this  from  the  investment  value 96.12 

We  obtain  a  new  investment  value  (as  above) $10,098.04 

The  nominal  interest  for  the  next  year  is $300.00 

The  effectiv  interest,  2%  of  $10,098.04,  is 201.96 

The  amortization  is .T7T7TT7  98.04 

Leaving  THE  Par $10,000.00 

Thus  the  results  by  compound  discount  and  by  gradual  amortiza- 
tion are  precisely  the  same;  and  in  either  case  the  effectiv  rate  is  actually 
received  and  the  amount  calculated  to  maturity  is  exactly  what  it  should 
be. 

Sprague's  Extended  Bqnd  Tabi^es  (2nd  Edition,  New  York, 
1907)  are  accurately  calculated  on  precisely  the  above  principle.  All 
,values  are  given  on  the  assumption  of  half-yearly  conversion,  which 
corresponds  with  the  serai-annual  periods  covered  by  the  reports  of 


Amortization  8 

savings  banks.  Their  use  in  computing  schedules  of  amortization  (which 
was  the  original  purpose  in  constructing  them)  will  save  many  tedious 
hours  of  computation,  and  their  extreme  minuteness  (to  the  nearest  cent 
on  one  million  dollars),  as  well  as  their  wide  range  of  incomes  {l%%  to 
10%),  make  them  the  only  tables  which  will  fully  meet  this  need. 


PreIvIminary  Steps. 


As  the  reports  to  the  Banking  Department  show  the  condition  at 
the  opening  of  business  on  the  first  day  of  January  and  the  first  day  of 
July,  we  shall  assume  that  amortization  is  uniformly  to  be  effected  up  to 
the  close  of  business  on  the  last  days  of  December  and  of  June,  altho  the 
interest  may  be  payable  at  some  other  dates. 

An  exact  list  of  all  the  holdings  of  bonds  must  be  prepared,  giving 
all  the  data  necessary  for  the  future  account  on  the  investment  value 
basis.  This  may  be  made  up  in  the  tabular  form,  or  a  paper  or  ticket  may 
be  devoted  to  each  lot  of  bonds. 

The  following  are  the  data  to  be  obtained  from  the  books  as  to  each 
lot  of  bonds.  Where  bonds  of  a  certain  issue  were  purchased  in  several 
lots  at  different  times  and  prices,  they  must  be  treated  separately  at  this 
stage. 

1.  Thenar  or  principal  sum. 

2.  The  date  when  it  will  become  due. 

3.  The  rate  of  interest  stipulated  to  be  paid. 

4.  The  dates  of  interest  payments. 

5.  The  date  of  purchase. 

6.  The  price  per  $100  paid  and  the  total  cost,  exclusive  of  accrued 
interest. 

From  the  above  data  is  to  be  ascertained  the  income-basis  upon 
which  the  purchase  was  made,  and  to  which  all  subsequent  computa- 
tions must  conform. 


Determining  the  BAvSis. 

As  the  entire  computation  of  investment  values  depends  upon  the 
basis  of  income,  the  first  step  is  to  ascertain  what  was  the  basis  at  which 
each  lot  of  bonds  was  purchased. 

If,  at  the  time  6f  each  purchase  a  record  was  made^  showing  the 
basis  at  the  cost  price,  then  the  basis  may  be  at  once  inserted  in  the 
descriptiv  ticket  and  much  labor  saved.     But  this  is  not  always  the  case. 

It  must  be  remembered  that  after  finding  the  basis,  it  is  not  neces- 
sary to  continue  the  successiv  values  down  to  the  time  of  installing  the 
plan  of  amortization.  On  the  inaugural  date^  as  we  shall  call  it,  a  fresh 
start  is  made  with  the  inaugural  value,  and  the  values  listed  until  they 
reach  par  at  maturity. 

The  bond-tables  contain  in  the  left-hand  column  of  each  page  the 
basis  of  income  from  2.50%  to  5.00%,  between  which  almost  all  savings 


4  Amortization 

bank  investments  are  found.  Rates  above  and  below  these  are  given  in 
the  second  edition.  These  are  given  one-twentieth  of  one  per  cent,  apart, 
2.50,  2.55,  2.60,  2.65,  2.70,  etc.  Intermediate  values,  one  hundredth 
of  one  per  cent,  apart  (as  2.51,  2.52,  2.53,  2.54)  are  found  by  takmg  so 
many  fifths  of  the  difference  and  then  correcting  by  means  of  the  blue 
pages. 

We  will  first  explain  the  method  of  determining  the  basis,  finding 
the  inaugural  value  and  completing  the  amortization  to  maturity,  on  the 
assumption  that  the  tabular  values  (2.05,  2.10,  2.15,  2.20,  etc.)  are  the 
ones  which  apply  or  else  are  considered  near  enough  for  practical 
purposes. 

1.  For  Even  Hai^f-years. 

The  income  basis  depends  upon  three  things  :  the  cash  rate,  or 
coupon  rate  (as  a  3%  or  a  6%  bond),  the  price  (exclusi  v  of  accrued  interest), 
and  the  time  the  bond  has  to  run. 

Let  us  take  as  an  illustration  a  6%  bond  for  $20,000  which  was 
purchased  when  it  still  had  just  42  years  to  run,  at  132.88.  We  turn 
to  that  portion  of  the  tables  devoted  to  5%  bonds;  find  the  column 
headed  42  years,  and  run  down  the  column  until  we  come  to  the  nearest 
value  to  132.88.  We  find  that,  at  the  income  basis,  3.50,  the  value  of 
$100  would  be  132.877107,  hence  3.50  is  practically  the  exact  income 
basis  of  the  bond,  and  its  present  investment  values  must  be  computed 
on  that  basis. 

If,  at  the  inaugural  date  the  bond  still  has  12  years  to  run,  then 
turning  to  the  12  year  column  on  page  83  we  find,  opposit  the  net  income 
basis  3.50,  the  investment  value  of  $1,000,0C0  as  1,145,955.14,  and  con- 
sequently the  inaugural  value  for  $20,000  par  will  be  $22,919.10.  This 
value  will  furnish  3^%  income  on  the  amount  remaining  invested  and 
will  reduce  the  investment  to  par  on  the  day  of  maturity. 

Where  a  number  of  bond  accounts  are  reduced  at  the  same  time  to 
the  investment  value,  the  result  will  be  substantially  accurate  if  computed 
on  the  tabular  values,  to  the  nearest  1/20  of  one  per  cent.  We  will  here- 
after explain  the  more  minute  calculations,  by  which  accuracy  to  the 
1/100  of  one  per  cent,  may  be  attained. 

2.  For  Broken  Times. 

In  the  foregoing  examples,  it  has  been  supposed  that  the  purchase 
was  made  on  an  interest  date,  so  that  the  search  is  confined  to  a  single 
column.  It  is,  however,  very  frequently  the  case  that,  besides  the  years 
and  half  years,  there  are  odd  months  and  days  in  the  time  which  the 
bond  has  to  run.  In  this  case,  the  time  and  hence  the  value  must  lie 
between  two  columns  of  the  table,  and  the  difference  of  the  values  is  di- 
vided proportionately  to  the  difference  of  the  times. 

On  page  8  of  the  Banking  Department  circular,  authority  is  given 
for  treating  the  inaugural  value  as  of  the  precedmg  January  or  July  first 
aud  for  disregarding  any  terminal  portion  of  a  period,  thus  reducing  the 
computation  to  even  half-years.     All  inaugural  values  are  then  found  in 


Amortization  5 

the  tables.    To  extract  them  is  far  easikr  and  at  The  same  time 

FAR  MORE  accurate  THAN  THE    "  SIMPI^E  "   METHOD.      But  if  absolute 

accuracy  is  desired,  the  basis  may  be  found  as  follows  : 

If  the  given  value  lies  between  two  values,  and  no  others  of  the 

adjoining  columns,  the  income-basis  of  that  line  is  the  basis  sought, 

within  1/20  of  1%.     For  example,  a  6%  bond  having  18  years,  9  months 

and  6  days  to  run,  bought  at  140  and  interest;  what  is  the  basis?    The 

time  may  be  better  exprest  as  18>^  years,  3  months  and  6  days.     Each 

day  is  computed  as  the  30th  of  a  month  and  each  month  as  the  6th  of  a 

half  year. 

We  must  look  down  both  the  18^  year  column  and  the  19  year 

column,  to  see  if  we  can  find  a  pair  of  values  between  which  140  lies. 

We  find  the  pair:  iot/  in 

^  18;^  yr.  19  yr. 

3.15  1  397  278  83  1  405  147  76 

and  this  is  the  only  pair  which  embraces  140;  and,  therefore,  3.15  is  the 
income-basis  within  the  limits  of  1/20%;  no  other  tabular  value  will  do. 
But  if  there  are  two  pairs  between  either  of  which  the  value  for  the  given 
time  might  lie,  or  if  there  are  no  such  pairs,  we  shall  have  to  interpolate 
according  to  time. 

If  the  price  had  been  120  and  interest  and  the  time  8}4  years, 
3  months,  6  days,  we  should  have  to  choose  between  these  pairs  : 

S'A  yr.  9  yr. 

3.30  1  198  706  06  1  208  761  50 

3.35  1  194  614  08  1  204  439  71 

Either  pair  has  120  within  its  limits. 

The  difference  between 1208  76150 

and ,      1  198  706  06 

is  the  amortization  for  6  months 10  055  44 

three  months  will  be  half  as  much 5  027  72 

6  days  will  be  1/30  of  10,055-44  or 335  18 

3  months,  6  days,  will  be 5  362  90 

This  added  to 1  198  706  06 

gives  the  investment  value  at  3.30% 1  204  068  96 

1  204  439  71 
1  194  614  08 

9  825  63 

By  the  same  process  applied  to  the  rate  3  35 /  o.^J  ^9 

5  240  33 
1  194  614  08 

1  199  854  41 

we  find  the  much  nearer  approximate,  hence  3.35  is  the  proper  basis. 
It  is  true  even  to  the  1/100  of  1%,  as  3.30  appears  by  inspection  to  be 
wider  of  the  mark. 

If  advantage  be  taken  of  the  permission  given  by  the  Banking 
Department,  page  8,  all  purchases  and  all  redemptions  will  be  treated  as 
of  June  30th  and  December  31st.  This  will  be  easier,  but  will  give  a  less 
accurate  result. 


6  Amortization 

3.  For  Seriai.  or  Various  Maturities. 

Bonds  are  often  issued  in  series.  For  example:  $30,000,  of  which 
$1,000  is  payable  after  one  year,  another  $1,000  after  two  years,  and  the 
last  $1,000,  30  years  from  date. 

In  offering  such  bonds  for  sale,  they  are  often  listed  as  of  "average 
maturity — 15)4  years."  This  is  entirely  delusiv,  and  frequently  causes 
the  buyer  to  believe  that  he  is  getting  a  more  favorable  basis  than  will 
be  realized.  The  only  correct  valuation  of  a  series  is  the  sum  of  all  its 
separate  values.  If  we  assume  that  the  $30,000  above  referred  to  was  a 
series  of  5%   bonds  bought  on  an  alleged  3.50%  basis,  the  true  value 

would  be 35,005.00 

whereas  the  value  for  the  *'  average  time  "  would  be 35,348.22 

In  computing  the  basis  of  a  series,  the  basis  corresponding  to  the 
average  time  may  be  taken  as  a  point  of  departure,  but  it  will  be  found 
that  it  is  invariably  too  high. 

A  series  of  30  5%  bonds  maturing  at  1  to  30  years  from  the  date  of 
purchase  at  116.68;  what  basis  ? 
looking  in  the  15)4  year  column    (average    date)    the 

nearest  value  to  116.68  is 1  165  200  00 

which  is  at  a  3.60  basis. 

If,  however,  we  take  off  on  an  adding  machine  the  values 

for  30  years  at  3.60 1  255  549  38 

for  29  years  at  3.60 1  250  705  95 

and  so  on  to  one  year,  we  shall  find  that  the  total  is 34  531  390  28 

and  the  average  price 115.10 

The  basis  3.55  will  give  the  result  116.06,  but  3.50  will  give  116.68,  and 
3.50  is  the  true  basis  on  separate  maturities. 

4.  Bonds  at  Unusuai.  Rates. 

All  the  usual  rates  (2,  3,  3>^,  4,  4)4,  5,  6  and  7  per  cent  bonds)  are 
provided  for  in  the  tables.  Infrequently,  bonds  of  different  rates  occur, 
as  3.60,  3.65,  3^,  4Xi  5^.  These  may  accurately  and  readily  be  de- 
rived from  the  tabular  values  by  "splitting  the  difference  " :  4)(  bonds 
are  midway  between  4's  and  4j^'s  of  the  same  income  basis  :  5>^'s 
between  5  and  6.  For  3.60  bonds,  to  the  value  of  a  3^  add  1/5  the  differ- 
ence between  S^  and  4. 

For  example,  a  3%  bond  of  25  years  at  a  3 .  55  basis 

is  worth 909  350  85 

and  a  4%  bond  on  the  same  basis  is  worth 1  074  167  48 

The  difference  is 164  816  63 

We  should,  therefore,  expect  a  5%  bond  to  be  worth 1  238  984  11 

which  is  found  to  be  true  on  turning  to  page  88.  The  dif- 
ference for  y2%  should  be  half  of  164  816  63,  or 82  408  31,5 

and  the  value  of  a  4yi/c  bond  on  a  3.55  basis  is 1  156  575  79,5 

A  3%  bond  is  worth 909  350  85 

4-     82  408  31.5 

And  a  3>^%  bond  is  worth 991  759  16,5 

Adding  to  this  .15  of  164  816  63 +  24  722  49.45 

we  have  the  value  of  a  3  65  bond 1  016  481  66 


Amortization  '  7 

The  Inaugural  Valuk. 

Assuming  that  the  entire  system  of  amortization  is  to  be  inaugu- 
rated as  of  a  certain  date,  which  we  call  the  inaugural  date,  and  that  the 
income  basis  has  been  ascertained,  it  remains  to  compute  the  value  on 
the  inaugural  date. 

In  our  illustrations  we  shall  assume  the  1st  of  January,  1909,  as 
the  inaugural  date. 

The  process  of  finding  the  value  is  just  the  reverse  of  that  for 
finding  the  basis,  but  it  is  done  with  greater  nicety,  to  the  last  decimal. 

The  first  step  is  to  compute  the  time,  that  is,  the  number  of  years, 
months  and  days  from  January  1st,  1909,  to  the  day  of  maturity,  and  this 
time  is  also  to  be  entered  on  the  list  of  data. 

All  the  computations  should,  if  possible,  be  made  on  blank  books 
rather  than  on  loose  pieces  of  paper.  Paper  ruled  like  the  inside  cover 
of  this  book  is  recommended,  to  be  made  up  into  manila-covered  books 
of  convenient  size,  say  about  7  x  10  inches,  paged  consecutively  from 
book  to  book,  for  reference. 

Each  inaugural  value  will  be  computed  on  this  book,  either 
followed  by  the  successiv  investment  values  down  to  maturity  or  by 
space  enough  to  contain  them  if  that  work  is  postponed. 

1.  For  J.  AND  J.  Bonds. 

If  the  income-basis  is  one  of  those  given  in  the  body  of  the  tables, 
it  is  only  necessary  to  turn  the  appropriate  column  and  multiply  the 
value  there  found  by  the  amount  at  par,  to  as  many  figures  as  required. 

2.  For  Bonds  at  other  Dates. 

Assuming  that  advantage  is  not  taken  of  the  authority  on  page  8 
of  the  Banking  Department  Circular,  the  procedure  is  as  follows  : 

Copy  the  values  from  the  column  earlier  and  the  column  later 
than  the  time  given,  and  find  the  difference. 

This  difference  is  the  amortization  (or  "run-off")  for  C  months; 
for  one  month  it  must  be  ]/(,  of  this  difference  and  for  other  numbers  of 
months,  various  fractions  as  usually  computed. 

For  F.  and  A.  bonds  Ye  of  the  difference  ; 

For  M.  and  S.  >^  ; 

For  A.  and  O.  Yz  ; 

For  M.  and  N.  2^,  and 

For  J.  and  D.  5/6  of  the  difference  are  to  be  added  to  the  shorter 
time  value,  for  bonds  above  par,  and  subtracted  from  the  shorter  time 
value  for  bonds  below  par. 

If  among  our  5%  bonds  we  have  six  different  lots,  $10,000  each,  all 
on  a  3.80  basis,  and  maturing  as  follows  : 

(a)  January  1st,  1919,  J.  and  J.,  10  years. 

(b)  February  1st,  1919,  F.  and  A.,  10  years,  1  month. 

(c)  March  Ist,  1919,  M.  and  S.,  10  years,  2  months. 

(d)  April  1st,  1919,  A.  and  O.,  10  years,  3  months. 

(e)  May  1st,  1919,  M.  and  N.,  10  years,  4  months. 

(f)  June  1st,  1919,  J.  and  D.,  10  years,  5  months  : 


8  Amortization 

Turning  to  page  82,  we  find  that  the  value  of  lot  (a)  is  $10,990.62. 

The  others  must  be  a  little  larger,  lying  between 10,990 .  6200 

and  10;^  years 11.031.0304 

Difference  for  six  months 40 .4104 

Of  this  difference  we  take  the  following  fractions: 

Ye  is 6.7351 

yi  is 13.4701 

yi  is 20.2052 

2^  is 26 .  9403 

5/6  is 33.6753 

Values  rounded  off  to  the  nearest  cent :  of  the 

February  bond 10,990  6200  +     6 .  7351  =  10,997 . 36 

March  bond 10,990.0200  +  13.4701  =  11,004.09 

April  bond 10,990.6200  +  20.2052  =  11,010.83 

May  bond 10,990.6200  +  26.9403  =  11,017.56 

June  bond 10,990.6200  +  33.6753  =  11,024.30 

Bonds  having  interest  payable  on  other  days  of  the  month  are 
similarly  prorated  by  months  and  days,  each  interest-day  being  treated 
as  the  180th  part  of  a  half  year. 

It  may  be  noted  that  in  May  and  June  it  would  have  been  easier  to 
find  Yi  and  Ye  than  %  ^^^  ^/^I  ^^^  ^^^  same  result  would  have  been  ob- 
tained by  subtracting  Y  or  Ye  from  the  greater  value  as  in  adding  %  or 
5/6  to  the  less  value. 

11,031.0304  —  13.4701  =  11,017.56 
11,031.0304  —    6.7351  =  11,024.29 

3.    Seriai.  Bonds. 

The  inaugural  value  of  a  series  of  bonds  is,  of  course,  the  sum  of  all 
the  values  at  the  proper  times  and  basis.  As  already  shown,  the  readiest 
method  is  to  transcribe  these  values  on  the  adding-machine. 

The  case  of  a  J.  and  J.  bond  is  easiest.  Suppose  on  the  inaugural 
date  we  have  a  series  of  5%  bonds,  of  |10,000  each,  maturing  July  1st, 
1915  to  1919,  incl.,  and  that  the  basis  is  3.60.  The  values  are  as  follows 
for  11,000,000  : 

Maturing  July  1st, 


1915 

6>^  yrs. 

1  080  495  95 

1916 

7>4  yrs. 

1  091  305  38 

1917 

8>^  yrs. 

1  101  735  92 

1918 

9K  yrs. 

1  111  800  87 

1919 

10>^  yrs. 

1  121  513  03 

5  506  851  15 

The  value  of  the  five  bonds  will,  therefore,  be  $55,068 .51.  This  is 
the  easiest  way  of  finding  the  inaugural  value  ;  but  with  a  view  to  the 
future  values,  it  is  recommended  to  commence  at  yi  year  and  continue 
the  addition  to  10>^  years,  the  longest  time,  with  a  sub-total  after  each 
value. 


Amortization  9 

K  yr 1  006  876  23 

lYz   yrs 1  020  266  08 

2  027  142  31 
2K  yrs 1  033  186  61 

3  060  328  92 
3K  yrs 1  045  654  27 

4  105  983  19 
4^2  yrs 1  057  684  92 

5  163  668  11 
hYz   yrs 1  069  293  89 

6  232  962  00 
6>^  yrs 1  080  495  95 

7  313  457  95 
lyi   yrs 1  091  305  38 

8  404  763  33 
^Yz   yrs 1  101  735  92 

9  506  499  25 
9>^  yrs 1  111  800  87 

10  618  300  12 
10%   yrs. ..... .  1  121  513  03 

11  739  813  15 

By  subtracting  from  this  total  all  the  values  preceding  that  for  6>^ 
years  we  have  the  inaugural  value  as  before: 

11  739  813  15 
6  232  962  00 
5  506  851  15 

If  the  bonds  are  not  J.  and  J.,  but  some  other  date,  it  will  be  neces- 
sary to  take  off  the  values  for  1  year,  2  years,  3  years,  etc.  This  will 
cause  no  loss  of  time,  as  those  values  would  anyhow  have  to  be  taken  off 
for  future  amortization : 

1  yr 1  013  630  87 

2  yrs 1  026  783  97 

2  0^0  414  84 

3  yrs ]  039  476  04 

3  079  890  88 

4  yrs 1  051  723  25 

4  131  614  13 

5  yrs 1  063  541  18 

5  195  155  31 

6  yrs 1  074  944  88 

6  270  100  19 

7  yrs 1  085  948  87 

7  356  049  06 

8  yrs 1  096  567  17 

8  452  616  23 

9  yrs 1  106  813  28 

9  559  429  51 
10  yrs 1  116  700  26 

10  676  129  77 

5  195  155  31 

6  480  974  46 


10  Amortization 

The  difference  between 5  506  851  15 

and 5  480  974  46 

which  is 25  876  69 

measures  6  months  added  to  the  life  of  the  bonds.    If  they 

matured  in  Feb.  instead  of  July,  ye  of  25  876  69  being. . .  4  312  78 

must  be  added  to 5  480  974  46 

giving  the  value 5  485  287  24 

and  similarly  for  March 5  489  6U0  02 

"     April 5  493  912  80 

♦•     May 5  498  225  59 

♦•    June 5  502  538  37 


Combining  Severai,  Purchases. 

When  a  certain  security  has  been  invested  in  at  various  times,  vari- 
ous quantities,  various  prices,  and  various  income  bases,  the  basis  of  each 
lot  must  be  separately  ascertained,  as  has  been  explained.  But  it  is  not 
necessary  to  carry  as  many  accounts  as  there  are  lots  ;  a  combined  value 
may  be  ascertained  and  an  equated  income  basis,  upon  which  the  amorti- 
zation will  proceed,  giving  a  joint  result  substantially  the  same  as  the 
sum  of  the  separate  values,  and  with  less  labor. 

We  have  |22,000  5%  bonds  of  a  certain  issue,  due  January  1st, 
1920,  purchased  as  follows : 

$10,000  Jan.  1,  1900,  for  $12,992.00,  a  3%  basis. 
5,000  Jan.  1,  1903,  for     5,955.00,  a  3>^%  basis. 
7,000  Jan.  1,  1905,  for      7,340.00,  a  4.55%  basis. 

$22,000  cost    $26,287.00 

Each  parcel  is  to  be  reduced  to  its  value,  at  3%,  3>^%  and  4.55% 
respectivly,  on  the  inaugural  date,  January  1st,  1908. 

$10,000  on  a  3%  basis  =  $12,003.04 
5.000  on  a  3J^%  basis  =  5,729.77 
7,000  on   a  4.55%   basis    ^      7,288.82 

Investment  value  Jan.  1,  1908  $25,021.63 

At  the  nearest  basis  3.58%,  the  bonds  are  valued  at  $25,025.64, 

which  we  adopt  as  the  inaugural  value,  and  this  will  amortize  the  bonds 

to  par  at  maturity.     The  basis  is  actually  3.5818,  or  $25,021.63. 

It  is,  therefore,  unnecessary  to  treat  each  lot  of  bonds  separately  in 

the  scientific  method  of  amortization. 

The  remarks  of  the  Superintendent  on  page  10  of  his  circular  seem 

to  refer  to  the  pro  rata  method. 


Continuing  the  Amortization. 

Having  establisht  the  inaugural  value,  we  must  find  successiv  in- 
vestment-values all  the  way  down  to  par,  and  the  differences  between 
these  values  are  the  amortization  or  the  accretion  as  the  case  may  be. 

The  first  question  which  arises  is  this:  shall  we  (1)  calculate  these 
on  each  bond  all  the  way  to  maturity,  making  a  complete  schedule? 
or  (2)  shall  we  only  continue  each  class  of  bonds  for  one  period  and  per- 
form this  work  every  half  year  ? 


Amortization  11 

If  there  is  enough  time,  I  would  recommend  plan  (1),  filling  up  a 
complete  schedule  to  maturity.  If  the  premium  or  discount  has  then 
exactly  disappeared  it  is  the  strongest  evidence  that  all  the  successiv 
values  are  correct.  It  is  evidently  easier  to  keep  at  work  on  one  process 
than  to  shift  from  one  to  another,  and,  therefore,  the  plan  of  finishing 
one  before  taking  up  another  is  labor  saving. 

If,  however,  the  time  allowed  for  transforming  the  accounts  is  very 
short,  it  may  be  necessary  to  compute  only  one,  or  a  very  few,  of  the 
values  at  present,  deferring  the  completion  of  the  schedule.  In  this  case, 
space  should  be  left  in  the  calculation-books  for  continuing  the  computa- 
tions to  maturity. 

There  are  two  general  methods  for  amortization  (or  accretion): 

1.  By  transcription  ; 

2.  By  multiplication. 

The  former  consists  in  deriving  each  value  independently  from  the 
tables;  the  latter  in  deriving  each  value  from  its  predecessor,  which  is 
multiplied  by  the  income- rate,  the  cash  received  being  then  subtracted. 

We  will  exemplify  both  of  these  methods  and  show  when  they  are 
applicable,  respectively. 

1.    On  J.  AND  J.  Bonds. 

Here  the  values  for  $1.,  110.,  $100.,  $1,000.,  $10,000.,  $100,000.  or 
$1,000,000.  are  at  tabular  values;  that  is,  they  are  found  in  the  columns 
of  the  book  of  tables  and  require  only  to  be  multiplied  by  such  number 
as  will  correspond  to  the  principal  in  question. 

Thus,  we  have  determined  that  3.70  is  the  true  income  basis 
for  $30,000.   5%   bonds   due  January  1st,    1914,    the    inaugural    value 

being $31,765 .43 

Write  down  the  values  for  five  years,  four  years,  etc.,  of  $10,000, 
pointing  off  from  the  table,  but  retaining  the  mills;  opposit  each  value  of 
$10,000,  multiply  it  by  3,  to  cents;  the  carrying  figure  from  the  mills 
being  added  in : 

5yrs.         10,588.47,7    X    3    31,705.43 
AYi  yrs.     10,534.36,4  81,603.09 

4yrs.        10,479.24,9  31,437.75 

3>^  yrs.     10,423.11,6  31,269.36 

3  yrs.        10,365.94,3  31,097.83 

2>^yrs.     10,307.71,3  30,923.14 

2  yrs.        10,248.40,6  30,745.22 

l>^yrs.     10,188.00,1  30,564.00 

1  yr.  10,126.48,0  30,379.44 

>^yr.        10,063.81,9  30,191.46 

The  figures  on  the  right  are  reliable  as  the  successiv  investment- 
values;  the  only  possible  error  would  be  from  having  incorrectly  set  down 
the  first  column  or  from  having  made  an  error  in  multiplying  by  3. 

Both  of  these  may  be  effectually  checkt  by  the  use  of  an  adding- 
and-listing  machine.  Let  a  list  of  the  right-hand  column  be  made,  with 
total;  also  a  list  of  values  of  $1,000,000.  on  which  they  are  based,  with  total, 


12 


Amortization 


which  is 10,332,556.86 

Multiplying  by  3 30,997,670.58 

which  indicates  that  the  total  of  the  other  column  should  be     309,976 .  71 

In  fact  it  is 309,976.72 

and  the  trifling  discrepancy  may  be  ignored.     Small  differences  in  the 
cents  are  usually  caused  thru  the  suppression  of  decimals. 

This,  in  such  a  case,  is  of  course  the  easiest  way  of  obtaining  the 
values  required.  In  making  up  the  schedule  the  values  occupy  the  right- 
hand  column;  the  column  preceding  it  is  derived  by  subtraction;  the  one 
before  that  again  by  subtraction  from  the  interest  column,  which  is 
constant.  

ScHKDuivE  OF  Amortization. 


Date 

Interest 

Net  Income 

Amortization 

Investment 
Value 

1909  Jan.   1 
July  1 

1910  Jan.   1 
July  1 

1911  Jan.   1 
July  1 

1912  Jan.  1 
July  1 

1913  Jan.   1 
July  1 

1914  Jan.   1 

750  66 
750  00 
750  00 
750  00 
750  00 
750  00 
750  00 
750  00 
750  00 
750  00 

587*66 
584  66 
581  61 
578  47 
575  31 
572  08 
568  78 
565  44 
562  02 
558  54 

162 '34 

165  34 

168  39 
171  53 
174  69 
177  92 
181  22 
184  56 
187  98 
191  46 

31.765  43 
31,603  09 
31,437  75 
31,269  36 
31,097  83 
30,923  14 
30,745  22 
30,564  00 
30,379  44 
30,191  46 
30,000  00 

7,500  00 

5,734  57 

1,765  43 

The  totals  furnish  several  valuable  checks  on  the  accuracy  of  the 
schedule. 

The  total  amortization  must  equal  the  inaugural  premium,  or  the 
total  accretion  must  equal  the  inaugural  discount :  31,765.43  —  30,000  = 
1765.43. 

The  total  amortization  equals  the  total  interest  less  the  total  net 
income  :  7500  —  5734.57  =  1765.43. 

The  several  amortizations  bear  the  following  relation  to  one  an- 
other: each  is  the  product  of  the  preceding  by  1 .0185;  or  each  equals  the 

70%. 


preceding  -{-  six  months  interest  at  3. 

162.34    X  1.0185 
1.62 
1.30 

.08 


165.34 

1.65 

1.32 

.08 

168.39  etc. 


Amortization  13 

The  amortization  of  the  above  premiums  by  multiplication  would 
be  as  follows  : 

Inaugural  value  (retaining  the  mills) 31,765.43,1 

X  1.0185:    viz:     1 31,765.43,1 

.01 317.65,4 

.008 254.12,3 

.0005 15.88.3 

32,353.09,1 
—  750 750.00 

31,603.09,1 

316.03,1 

252.82,4 

15.80,2 

32.187.74,8 
750.00 

31,437.74,8 
and  so  on  to  maturity. 

If  the  mills  figure  had  not  been  retained  there  would  have  been 
some  inaccuracy  in  the  last  figure. 

2.    Intkrest-days  other  than  January  1st  and  Jui.y  1st. 

The  successiv  values  for  intermediate  dates  are  computed  in  the 
same  way  as  the  inaugural  value  by  transcription  of  the  tabular  values 
and  interpolation  by  simple  proportion.  The  following  procedure  will 
facilitate  and  check  these  operations.  We  will  illustrate  it  by  the  follow- 
ing problems  : 

100,000  4%'s  on  a  3.50  basis,  and 
100,000  4%'s  on  a  4.50  basis, 

payable  February  1st,  1912;  3  yrs.,  1  mo. 

Set  down  the  values  for  even  years  and  half  years,  beginning  with 
S)4  years,  leaving  spaces  between  the  lines  and  on  each  side.  It  is  as 
well  to  set  down  the  extended  figures  for  the  full  |1, 000,000.  First  for 
the  3.50  basis  : 

1  016  336  60 

1  014  122  49 
1  Oil  869  64 
1  009  577  36 
1  007  244  96 
1  004  871  75 
1  002  457  00 
1  000  000  00 


14  Amortization 

These  amounts,  exclusiv  of  the  first,  are  totaled  on  the  machine 
for  the  purpose  of  proving  :     ^  q^q  ^^^  go 

But  the  millions  may  be  neglected,  leaving 

50  143  20 
Next  subtract  each  value  from  the  next  above  and  place  the  differ- 
ence on  the  left  of  the  lower;  total  these  differences 

1  016  336  60 
2  214  11  1  014  122  49 

2  252  85  1  Oil  869  64 

2  292  28  1  009  577  36 

2  332  40  1  007  244  96 

2  373  21  1  004  871  75 

2  414  75  1  002  457  00 

2  457  00  1  000  000  00 


16  336  60  50  143  20 

The  column  of  difference  aggregates  the  same  as  the  previous  pre- 
mium, which  indicates  that  no  error  has  been  made. 

As  this  is  an  F.  A.  Bond  we  take  ye  of  each  diflference  and  place  it 
above  the  adjoining  value  and  add  it: 

1  016  336  60 

+  369  02 
2  214  11  1  014  122  49  1014  491  51 

375  47 
2  252  85  1  Oil  669  64  1  012  245  11 

382  05 
2  292  28  1  009  577  36  1  009  959  41 

388  73 
2  332  40  1  007  244  96  1  007  633  69 

395  54 
2  373  21  1  004  871  75  1  005  267  29 

402  46        « 
2  414  75  1  002  457  00  1  002  859  46 

409  50 
2  457  00  1  000  000  00  1  000  409  50 


6)  16  336  60 


52  865  97  52  865  97 

Errors  of  multiplication  or  subtraction  are  detected  by  adding  Ye 
of  the  first  column  to  the  total  of  the  second,  which  should  produce  the 
third. 


Amortization 


15 


The  bond  on  a  4 .  50  basis  could  give  the  following  results,  the  yi 
being  subtracted  downward,  not  added: 


2  139  43 

2  187  56 
2  236  78 
2  287  11 
2  338  57 
2  391  18 

2  444  99 

6)  16  025  62 


983  974  38 

—  356  57 
986  113  81 

364  59 
988  301  37 

372  80 
990  538  15 

381  18 
992  825  26 

389  76 
995  J 63  83 

398  53 
997  555  01 

407  50 
1  000  000  00 

6  950  497  43 
2  670  94 

6  947  826  49 


985  757  24 
987  936  78 
.  990  165  35 
992  444  07 
994  774  07 
997  156  48 
999  592  50 

6  947  826  49 


The  effects  of  adjustment  for  intermediate  months  may  be  sum- 
marized as  follows: 

^-  -^^  ^  1  Above  par  add  to  1      i      r       t,     .     ♦• 

j^    3    ^  I  Below  par  subtract  from  j^^^"^^^''^^^^^^^  ^^°^«- 

A.  O.   %     Midway  between,  add  or  subtract. 

M.  N.  >^  I  Above  par  subtract  from  1      i      r     i 

J    ^     y^  \  Below  par  add  to  j  ^^^^^  ^^^^  l°°g^^  ^^°^«- 

A  schedule  formed  from  the  last  example  would  read  : 

SCHEDUI.E  OF  Accretion. 


Date 

Interest 

Net  Income 

Accretion 

Interest 
Value 

1909  Jan.  1 

98  575  72 

July  1 

2  666  6o 

2  217  96 

217  96 

98  793  68 

1910  Jan.  1 

2  000  00 

2  222  85 

222  85 

99  016  53 

July  1 

2  000  00 

2  227  88 

227  88 

99  244  41 

1911  Jan.  1 

2  000  00 

2  233  00 

233  00 

99  477  41 

July  1 

2  000  00 

2  238  24 

238  24 

99  715  65 

1912  Jan.  1 

2  000  00 

2  243  60 

243  60 

99  959  25 

Feb.  1 

333  33 

374  08 

40  75 

1  00  000  00 

12  333  33 

13  757  61 

1  424  28 

The  foregoing  problems  might  have  been  workt  by  multiplication, 
the  only  difficulty  being  at  the  fractional  period  at  the  end,  one  month. 
In  exemplifying  this  we  will  make  the  further  simplification  of  indi- 


16  Amortization 

eating  subtraction  by  a  line  drawn  around  the  figures  of  cash  interest,  and 
performing  the  subtraction  and  the  addition  at  the  same  operation. 
At  the  rate  3 .50  it  is  easier  to  take 

1% 01 

divide  it  by  2.... 005 

and  divide  that  by  2 .0025 

.0175 
than  to  multiply  .01 

by  7 007 

and  by  5 0005 

$100,000  4%  bonds  3.50%  basis,  payable  February  1st,  1909.     F.  &  A. 

Inaugural  value  January  1st,  1009 101  449  15 

6  mouths' interest  at  .0175      .01     1014  49 

.005  507  25 

.0025 /^'^^  ^^ 

Subtract  6  months'  cash  interest \2j 

Value  July  1,  1909 101  224  51 

Coutmue  the  operation 1  012  25 

506  12 
253  06 
2' 


© 


January  1,  1910 100  995  94 

1  009  96 
504  98 
^252  49 
2' 


& 


July  1,  1910 100  763  37 

1  007  63 

503  82 
251  91 
2' 


© 


January  1,  1911 100  526  73 

1  005  27 
502  63 
251  32 

2' 


© 


July  1,  1911 100  285  95 

1  002  86 
501  43 
^250  71 
2' 


© 


January  1,  1912 100  040  95 

We  have  brought  the  value  down  to  the  last  full  period.  For  the 
broken  period  it  is  evident  that  $40.95  is  the  amortization,  and  this  is 
corroborated  by  the  fact  that  it  is  exactly  ^  of  the  last  6  months'  pre- 
mium in  the  4%  table,  3 .  50  basis,  $245 .  70.  It  might  be  obtained  by  mul- 
tiplication, with  this  peculiarity  that  ^  of  the  par  plus  the  entire  pre- 
mium is  multiplied  by  .0175  and  yi  of  the  cash  interest  is  subtracted. 


Amortization  17 


16  666  67 

40  95 


16  707  62 


100  040  95 

167  08 
83  54 
41  77 

100  333  34 
333  34 

100  000  00 
But  this  process  is  unnecessary.     If  the  last  January  or  July  pre- 
mium isy^,  y^,  Yz,  ^  or  5/6  of  the  last  tabular  premium,  the  work  may 
be  considered  accurate. 

The  amortization  of  the  value  98,575  72  at  4.50  basis  would  be  as 
follows  :  we  retain  the  mills. 


98  575  72,4 

.02 

1  971  51,4 

.002 

197  15,1 

.0005 

49  28.8 

i  of  .002) 

© 

98  793  67,7 

1  975  87,4 

197  58,7 

49  39.7 

99  016  53,5 

1  980  33,1 

198  03,3 

49  50,8 
© 

99  244  40,7 

1  984  88,8 

198  48,9 

^^   49  62,2 

© 

99  477  40.6 

1  989  54,8 

198  95.5 

^^  49  73,9 
© 

99  715  64.8 

1  994  31,3 

199  43.1 

^^  49  85,8 

© 

99  959  25,0 

100,000  —  99,959,25  =  40.75 

100,000  —  99,755.50  =  244.50 

244.50-^6   =  40.75 


18  Amortization 

3.    Seriai,  Bonds. 

If  the  serial  bonds  are  of  the  J.  &  J.  maturity  they  may  be  re- 
computed at  each  half-year  by  the  method  outlined  under  ''Inaugural 
Values."  Two  complete  lists  are  made  up  on  the  adding  machine,  one 
of  the  even  years,  the  other  of  the  intervening  half  years,  beginnmg  at 
Yz  year  and  ending  with  the  longest  term.  Subtotals  are  inserted  after 
every  value.  If  the  series  has  commenced  to  mature,  the  total  is  given 
at  sight;  if  some  of  the  shorter  maturities  have  not  yet  arrived,  the  total 
of  those  earlier  bonds  must  be  subtracted  from  the  total  which  embraces 
all  now  outstanding.  In  short,  each  successiv  value  is  computed  exactly 
as  the  inaugural  vahie  was. 

But,  except  for  J.  &  J.  bonds,  it  will  usually  be  found  easier  to 
multiply  down,  and  even  with  J.  &  J.  it  may  be  done.  We  take  as  an 
example  a  series  of  three  bonds  maturing  March  1st,  1911  to  1913;  6% 
bonds,  3.80  basis.    We  take  the  full  extent  of  the  figures  in  the  tables. 

The  inaugural  value  January  1st,  1909,  is 3  194  386  49 

Until  the  bonds  begin  to  mature  we  proceed  as  usual,  mul- 
tiplying by        .019         .01  31  943  86 

.009 28  749  48 

3  255  079  83 

6  months'  interest  earned  on  $3,000,000 (9) 

July  1st,  1909 3  165  079  83 

31  650  80 

28  485  72 


© 


Jan.  1,  1910 3  135  216  35 

31  352  16 
28  216  95 


© 


July  1,  1910 .3  104  785  46 

31  047  85 
27  943  07 


© 


Jan.  1,  1911 3  073  776  38 

Here  a  complication  arises: 
$1,000,000  of  the  principal  is  only  invested 
for  2  months,  3^  of  a  period.     We  therefore 

compute  it  for  interest  as 333  333  33 

and  the  remainder  at  full  value 2  073  776  38 

.019  on 2  407  109  71  24  071  10 

21  663  99 

3  119  511  47 

We  must  subtract  principal 1  000  000 

Interest  earned  thereon 10  000 

"     on  2.000.000 60  1070  000  00 


Julyl,  1911 2  049  511  47 

During  the  half  year  July— Dec.  we  proceed  as  usual 20  495  11 

18  445  60 


© 


Jan.  1,  1912 2  028  452  18 


AmormzaTion  19 

Brought  forward 2  028  452  18 

This  being  the  half-year  in  which  a  payment  is  made  we 
proceed  as  follows : 

333  333  33 
1  028  452  18 

Interest  on 1  361  785  51 13  617  86 

12  256  07 

1  000  000  2  054  326  11 

10  000  1  040  000  00 

30  000  

Julyl,  1912 1  014  326  11 

10  143  26 
9  128  93 


® 


3^ 

1  003  598  30 
Instead  of  working  this  down,  we  verify  its  accuracy  by  observing 
that  the  premium  is  exactly  y^  of  §10,794.90,  the  tabular  premium  for 
6  months. 

New  Purchases. 

Future  purchases  will  doubtless  be  made  for  the  most  part  on  some 
agreed  basis.  If  not,  their  basis  will  be  found  precisely  as  in  earlier 
purchases. 

The  new  purchase,  if  not  on  any  exact  basis,  requires  to  be  squared 
up  by  the  next  following  balancing-period,  so  that  the  amortization  may 
thereafter  proceed  with  regularity.  On  long  bonds,  this  squaring-up 
process  may  sometimes  involve  a  considerable  adjustment,  enough  to 
make  a  perceptible  jar;  while  the  same  adjustment  spread  over  many 
years,  as  in  inaugurating  the  system,  would  be  imperceptible. 

It  is,  therefore,  in  the  new  purchases  that  the  most  exactness  is  re- 
quired and  it  may  be  thought  desirable  to  compute  them  to  a  higher 
degree  of  nicety,  in  respect  to  using  a  basis  correct  to  the  nearest 
hundredth  of  1%  ( .0001 )  and  in  other  particulars. 

Intermediate  Rates. 

The  tables  give  values  5/100  of  1%  apart,  as  2.50,  2.55,  2.60,  2.65. 

The  intermediate  values,  at  2.51,  2.52,  2.53,  2.54,  —  2.56,  2.57, 
2.58,  2.59,  may  be  obtained  approximately  by  taking  the  values  at  two 
tabular  rates  and  dividing  their  difference  by  5.     Thus  a  6%  bond  for  25 

years  at  the  basis  4.60  is  worth 1  206  716  65 

and  at  4 .  65 1  198  321  45 

The  interval  is 8  395  20 

one-fifth  is 1  679  04 

and  if  this  is  subtracted  successivly  from 1  206  716  65 

we  have  the  values  for  4.61 1  205  037  61 

4.62 1  203  358  57 

4.03 1  201  679  53 

4.64 1  200  000  49 

and  again  4  .65 1  198  321  45 


20  Amortization 

For  a  small  lot  of  bonds,  these  values  would  be  accurate  to  the 
nearest  cent,  but  on  $20,000  par  the  error  would  be  perceptible.  All  these 
values  are  too  large;  the  corrections  to  be  subtracted  are  found  in  the  blue 
pages;  in  this  case,  p.  178.  According  to  that  page  the  "difference" 
for  a  G%  bond  25  years,  basis  between  4.60  and  4. 05,  isG.G4andthe 
next  below  it  is  6.56.  The  difference  between  these  two  differences  is 
called  the  "sub-difference."  Our  corrections,  according  to  the  rules,  on 
pages  122 — 123  are  as  follows  : 

For  4.61,  the  difference 6.64 

For  4.62,  1^  times  the  difference  less  1/10 
the  sub-difference  9.95 

For  4.63,  1]4.  times  the  difference  less  1/5 

the  sub-difference 9.94 

For  4.64,  the  difference  less  1/5  the  sub- 
difference 6.62 

Subtracting  these  from  the  approximate  values  we  have  the  exact 
values  : 

1  205  037  61  —  6.64  =  1  205  030  97  (4.61) 
1203  358  57  —  9.95  -=  1203  348  62  (4.62) 
1  201  679  53  —  9.94  =  1  201  669  59  (4.63) 
1200  000  49    —    6.62    =    1199  993  87    (4.64) 

If  the  approximate  values  were  used  in  multiplying  down  there 
would  always  be  an  excess  at  maturity  and  this  would  be  an  increasing 
quantity,  thus  vitiating  the  test  afforded  by  reaching  par  at  maturity. 
But  it  will  be  found  that  the  corrected  values  will  multiply  down  with 
accuracy.  The  values  at  intermediate  rates  should  be  workt  out  first, 
before  interpolating  for  broken  times. 

In  serial  bonds  at  intermediate  rates  the  adjustment  may  be  made 
in  a  lump  on  the  aggregate.  Thus,  if  a  series  of  6%  bonds,  three  in  num- 
ber, due  in  18,  19  and  20  years,  respectively,  at  a  4% 

basis,  is  worth 3  792  849  62 

and  at  a  4.05  basis 3  760  788  11 

a  4.03  basis  would  be  approximately 3  779  012  71 

To  correct  this  take  off  the  differences  and 
sub-differences  opposit 4 .00 

onpagel77 18  yrs 4.56         .04 

19yrs 4.97         .05 

20  yrs 5.39        .05 

14.92         li 
lYz  times  14.92,  less  1/5  of  .14 22.35 

3  778  990  36 


Amortization  21 

QuARTERiyY  Bonds. 

A  quarterly  bond  of  a  certain  bond-rate  is  worth  more  at  a  certain 
basis  than  a  semi-annual  bond  at  the  same  basis.  If  it  is  decided  to  ignore 
the  advantage  of  collecting  half  the  interest  in  advance,  as  permitted  by 
the  Superintendent,  no  very  great  error  can  arise  and  the  bond  will  work 
out  like  any  semi-annual  one. 

If,  however,  it  is  desired  to  make  the  values  perfectly  exact,  a  set 
of  multipliers  will  be  found  on  page  VII  (p.  184  first  edition),  by  which 
the  premiums  or  discounts  are  to  be  multiplied,  increasing  the  value  of 
the  bond  in  either  case.  It  will  be  the  simplest  way  to  calculate  each 
successiv  premium  by  multiplication,  and  for  this  purpose  a  little  table 
may  be  constructed  of  the  multiple  in  question,  from  1  to  9  (Problems 
and  Studies  p.  20,  21).  For  example:  for  a  3.90  basis  on  a  5%,  the  table 
will  be  as  follows,  all  figures  below  the  second  line  being  formed  by  ad- 
dition of  the  top  line  : 


1 

022 

052  1 

2 

044 

104  2 

3 

066 

156  .S 

4 

088 

208  4 

5 

110 

260  5 

C 

132 

312  6 

7 

154 

361  7 

8 

176 

416  8 

9 

198 

468  9 

220 

521  0 

Bonds  at  a  lower  income-basis  than  2.50  are  entered  in  the  table 
(Additional  Volume,  included  in  second  edition)  on  a  basis  of  quarterly 
collection.  These  are  almost  entirely  U.  S.  bonds,  which  are  all  quarterly, 

Annuai,  Bonds. 

These  do  not  occur  so  frequently  as  quarterly  bonds,  but  multi- 
pliers for  reducing  the  premiums  or  discounts  to  the  semi-annual  stand- 
ard are  given  in  the  second  edition. 

Having  establisht  the  annual  values,  the  amortization  at  the  half- 
year  may  be  considered  as  one-half  of  the  yearly,  unless  great  accuracy 
is  required. 

Residues. 

Where  great  accuracy  is  intended,  it  often  happens  that,  having 
establisht  an  income-basis  as  close  as  possible,  it  is  found  that  there  is  still 
a  residue  of  difference  which  should  affect  the  rate  of  income,  but  it  is 
undesirable  to  extend  it  beyond  the  second  decimal  of  1%. 

There  are  three  ways  of  disposing  of  such  residues  : 

1.  By  taking  it  all  at  once  out  of  the  first  period's  amortization,  or 
adding  it,  as  the  case  may  be. 


2J  Amortization 

2.  To  distribute  it  over  the  list  of  values  by  dividing  into  equal 
partSy  to  be  added  to  or  subtracted  from  each  value. 

3.  To  distribute  it  still  more  accurately  over  the  list  in  proportion 
to  the  premiums  or  discounts. 

Thus,  if  a  certain  purchase  of  $1,000,000  5%  5  year  bonds  is  made 
at  104.50,  and  we  find  that  the  basis  of  4%  is  104.49129.,  which  is  nearer 
than  the  3 .  99%  basis.     This  leaves  a  residue  of  |8 .  71  to  be  disposed  of. 

1.  Put  it  all  into  the  first  amortization,  making  it  $418.88,  instead 
of  1410 . 1 7.     All  the  subsequent  values  are  regular. 

2.  Divide  $8.71  into  9  parts  of  87  cents  and  one  of  88  cents,  which 
are  added  to  the  amortization. 

3.  Divide  $8.71  into  -^siris  proportionate  to  the  several  amortiza- 
tions. These  will  vary  from  80  cents  for  the  first,  to  95  for  the  last. 
This  will  make  a  still  closer  approximation. . 


AccRUKD  Interest. 


That  portion  of  the  half  year's  interest  which  has  been  earned  and 
accrued  {''■grown  on")  is  exactly  as  much  part  of  the  det  secured  by  the 
obligation  as  the  principal  itself.  It  is  a  frequent  practis  to  ignore  the 
earnings  until  the  date  of  collection  ;  this,  while  convenient,  is  not 
strictly  correct.  It  is  not  the  cash  received,  but  the  earning-power  which 
makes  the  asset ;  the  collection  of  the  cash  is  an  incident . 

If  all  the  items  of  interest  were  payable,  and  were  punctually  paid, 
on  the  last  days  of  December  and  June,  the  result  would  be  the  same  ; 
but  when,  as  usually  happens,  there  is  a  single  balancing  period  for 
bonds  of  all  sorts  of  dates,  a  correct  balance-sheet  cannot  be  given  with- 
out adjusting  everything— interest  and  amortization  down  to  that  date. 

It  has  been  attempted  to  facilitate  the  calculations  by  amortizing 
to  the  next  previous  coupon-date  only  ;  but  this  would  be  no  saving  of 
labor,  for  the  accrued  interest  for  the  unexpired  term  would  have  to  be 
computed  at  the  effectiv  rate,  resulting  in  practical  amortization. 


AMORTIZATION  '  23 


The  Proper  Basis  of  Bond  AccoirNXs  When  Held  for 
Investment. 

By  Charles  E.  SpraguE,  Ph.D. 

From  The  Annals  of  the  American  Academy  of  Political  and  Social  Science, 
September,  1907,  entitled:  "  Bonds  as  Investment  Securities." 

A  bond  is  a  complex  promise  to  pay; 

1.  A  certain  sum  of  money  at  a  future  time ;  this  is  known  as  the 
principal,  or  par. 

2.  Certain  smaller  sums,  proportionate  to  the  principal,  and  at  vari- 
ous earlier  times.  These  are  usually  known  as  the  interest,  but  as  they 
do  not  necessarily  correspond  to  the  true  rate  of  interest,  it  will  be  better 
to  speak  of  them  as  the  coupons. 

The  sale  of  a  bond  is  the  transfer  of  the  right  to  receive  these  vari- 
ous sums  at  the  stipulated  times.  They  are  never  worth  their  face,  or 
par,  until  these  times  arrive,  but  are  always  at  discount.  The  principal 
is  never  worth  its  face  until  its  maturity  ;  the  coupons  are  never  worth 
their  face  until  their  maturities.  Yet  while  both  principal  and  coupons 
are  always  at  a  discount,  the  aggregate  may  easily  be  worth  more  than 
the  principal  alone  ;  and  it  is  the  aggregate,  principal  and  coupons,  which 
is  the  subject  of  the  bargain. 

The  purchaser,  in  fixing  the  price  which  he  is  willing  to  pay,  is 
guided  by  several  considerations  : 

1.  The  amount  of  the  principal. 

2.  The  amount  of  each  coupon. 

3.  The  length  of  time  to  which  the  principal  is  deferred. 

4.  The  number  of  coupons. 

5.  The  times  of  their  payments. 

6.  The  rate  of  interest  which  can  be  earned  upon  securities  of  a 
similar  grade. 

He  discounts  the  principal  and  each  coupon  at  compound  interest, 
at  such  rate  and  for  the  times  which  they  respectivly  have  to  run,  and  the 
sum  of  these  partial  present-worths  is  the  value  of  the  bond.  If  he  can 
buy  at  a  price  below  this  value  he  will  receive  a  higher  rate  of  interest 
than  he  anticipated  ;  if  he  is  required  to  pay  more  than  his  price,  he 
refuses  to  buy. 

As  he  cashes  each  coupon,  he  receives  what  he  paid  for  it  plus 
interest  at  the  uniform  rate  ;  thenceforward  he  earns  interest  on  a 
diminisht  investment  so  far  as  coupons  are  concerned,  but  on  an  increast 
investment  as  to  principal.  If  each  coupon  is  less  than  the  total  earning 
during  its  period  there  is  an  increase  in  the  total  investment ;  if  it  is 
greater,  there  is  a  surplus  which  operates  to  reduce  the  investment  or  to 
amortize  the  premium. 


24  Amortization 

We  have  then  two  fixt  points  in  the  history  of  the  bond  :  the  origi- 
nal cost  or  money  invested,  and  the  principal  sum  or  par,  or  money  to  be 
received  at  maturity.  Between  these  two  points  there  is  a  gradual 
change  ;  if  bought  below  par,  the  bond  must  rise  to  par ;  if  above,  it 
must  sink  to  par  ;  these  changes  being  the  effect  of  interest  earned  and 
coupons  paid.  At  any  intermediate  moment  there  is  an  investment-value 
which  can  be  calculated,  and  which  is  just  as  true  as  the  original  cost 
and  the  par.  In  fact  these  latter  are  merely  cases  of  investment- value ; 
the  in  vestment- value  at  the  date  of  purchase  is  cost ;  the  investment- 
value  at  the  date  of  maturity  is  par. 

The  gradual  change  in  the  investment  is  ignored  by  some  invest- 
ors, who  either  use  the  original  cost  all  thru,  or  the  par.  In  the  former 
case  they  suppose  that  the  investment  value  remains  at  its  original  figure 
until  the  very  day  of  maturity  and  is  instantly  reduced  to  par,  by  a  loss 
of  all  the  premium  or  a  sudden  gain  of  the  discount.  Those  who  use 
par  as  the  investment-value  assume  also  that  there  is  this  sudden  change 
of  value,  but  that  it  took  place  at  the  instant  of  purchase. 

These  treatments  are  manifestly  fictitious  and  unreal  and  only 
resorted  to  because  the  labor  of  computing  intermediate  values  is  shunned. 
Experience  would  tell  us,  if  theory  did  not,  that  there  is  no  such  violent 
change.  In  any  complete  system  of  bookkeeping  (popularly  called 
double  entry)  the  accounts  representing  assets  and  those  representing 
profits  and  losses  are  mutually  dependent.  You  cannot  arbitrarily  change 
a  value  without  affecting  and  distorting  the  general  profit-and-loss 
account.  A  year's  actual  gain  might  be  swept  away,  on  paper,  by  the 
investment,  perhaps  a  very  advantageous  one,  in  a  security  at  a  premium. 

The  disappearance  of  premium  being  regarded  as  a  consumption 
of  capital,  instead  of  a  return  requiring  reinvestment,  the  entire  coupon 
is  looked  upon  as  income  and  the  impairment  of  capital  becomes  actual. 
In  case  of  a  sale,  the  true  profit  or  loss  is  unknown,  the  proceeds  being 
compared  either  with  a  value  which  has  passed  into  history  or  one  which 
is  yet  to  be  realized — not  with  a  value  which  is  adjusted  to  the  present. 
The  error  in  these  faulty  methods  of  accountancy  arises  from  the  assump- 
tion that  interest  is  only  earned  when  specifically  collected  in  cash — that 
the  coupon  is  exactly  the  measure  of  the  interest  earned. 

When  the  bond  is  at  par  this  is  true  :  the  coupon  and  the  interest 
are  co-extensiv.  But  if  there  is  any  premium  or  discount  we  must  disre- 
gard the  distinction  between  principal  and  interest  and  consider  that  the 
original  investment  goes  on  increasing  at  compound  interest,  period  by 
period,  but  diminisht  by  the  coupons  and  the  final  redemption.  In  other 
words,  we  must  think  of  the  coupons  and  the  principal  as  merely  instal- 
ments, the  periodic  instalments  and  the  final  one,  but  all  of  the  same 
nature. 

A  familiar  instance  of  interest  earnings  not  represented  by  specific 
cash  payment  but  by  accretion  is  the  discount  of  a  note.  If  a  three- 
months' note  for  $1000  is  discounted  at  six  per  cent,  the  investment  is 
1985,  which  by  accretion  becomes  |1000.   Altho  interest  is  not  mentioned, 


Amortization  26 

the  purchaser  earns  $15,  or  more  than  6  per  cent,  on  his  investment  of 
$985.  If  the  note  were  payable  three  years  from  now  instead  of  three 
months,  he  would  expect  to  earn  compound  interest  and  would  pay,  per- 
haps, |837.48.  His  earnings  on  this  investment,  compounded  semi- 
annually, would  bring  the  investment  up  to  ^1000  in  three  years.  This 
note  would  be  equivalent  to  a  bond  without  coupons  ;  no  interest  is  stip- 
ulated for,  but  interest  is  actually  earned.  If  coupons  were  added,  the 
bond  would  simply  be  worth  so  much  more,  according  to  their  value. 

I  therefore  regard  the  cost  and  the  par  value,  while  correct  at  the 
beginning  and  at  the  end  of  the  period  of  ownership,  as  entirely  incorrect 
during  the  interim.  The  true  standard  is  the  present  worth,  compound- 
discounted,  of  all  recipiendSy  or  sums  of  cash  to  be  received,  whether 
called  coupons  or  principal. 

These  three  values  resemble  three  tenses  in  grammar  :  the  cost  is 
the  past,  what  was  paid  ;  the  par  is  future,  what  will  be  received ;  the 
investment  value  is  the  present.  There  is  a  fourth  value,  which  may  be 
considered  as  in  the  potential  mood  ;  what  might  be  obtained  on  sale,  at 
the  present  time.  This  is  the  market  value  and  is  a  matter  of  judgment, 
opinion  and  inference.  Some  bonds  are  bought  and  sold  so  frequently 
that  there  is  a  current  quotation  which  is  fairly  reliable ;  other  issues, 
in  which  dealings  are  rarer,  are  valued  by  analogy  with  those  whose  con- 
ditions are  nearest  like  them. 

It  may  be  observed  here  that  the  market  value  depends  solely  on 
the  rate  of  interest  which  prevails  on  the  particular  grade  of  security. 
This  has  sometimes  been  doubted  ;  even  the  courts  have  sometimes 
assumed  that  there  could  be  a  depreciation,  regardless  of  interest-rate, 
for  the  "badness"  of  the  investment;  or  a  premium  paid,  regardless  of 
interest-rate,  for  the  "excellence"  of  the  security.  It  is  necessary  to 
analyze  this  view,  which  I  regard  as  essentially  unsound. 

No  one  buys  a  bond  by  reason  of  admiration,  as  he  would  buy  a 
painting  or  a  statue.  He  is  dealing  solely  in  earnings,  that  is  to  say,  in 
interest.  He  is  impelled  by  no  other  motiv  than  that  of  receiving  his 
money  back  with  the  increment  which  shall  accru  in  the  meantime.  If 
an  investment  is  offered  him  which  is  superlativly  "good,"  but  which 
returns  only  what  he  originally  invested  without  any  increment,  he  will 
certainly  refuse  it. 

The  rate  of  interest,  however,  is  affected  by  the  risk  of  loss.  Every 
rate  of  interest  may  be  regarded  as  composed  of  two  parts :  one,  a  com- 
pensation for  the  use  of  the  capital ;  the  other,  a  premium  of  insurance 
against  chances  of  loss.  Thus  an  interest-rate  of  5  per  cent,  per  annum 
may  be  conceived  as 

3%  riskless  interest  or  compensation  for  use  of  capital ; 

+  2%  premium  for  insurance  against  loss. 
Another  and  safer  investment,  where  the  chance  of  loss  is  twice  as  remote 
would  rate  at  4  per  cent. : 

3%  riskless  interest ;  , 

-f-  1%  premium  of  insurance.  • 


<26  Amortization 

The  chance  of  loss  may  be  very  remote,  it  may  be  imaginary,  but 
it  is  worth  insuring  against.  Similarly,  the  chance  of  any  one  house 
being  destroyed  by  fire  is  remote,  but  men  willingly  pay  a  part  of  the 
income  of  the  house  to  secure  themselves  against  it.  The  loss  feared  in 
the  case  of  investment  is  not  merely  direct  loss,  or  failure  to  return,  but 
losses  by  delay,  by  difficulty  of  collection,  by  expense  of  litigation,  by 
the  very  feeling  of  suspense  which  acts  as  a  penalty.  An  opinion  that  the 
loss  is  possible  is  exactly  as  potent  as  the  reality,  in  producing  a  loading 
of  the  rate  on  account  of  risk-insurance,  provided  this  opinion  is  suffi- 
ciently widespread. 

A  lowering  of  the  grade  of  security  means  an  increase  of  the 
insurance-premium  and  hence  of  the  rate  of  interest.  This  may  happen 
by  deterioration  of  the  physical  property  which  underlies  the  investment, 
by  bad  management,  by  accidental  loss  of  custom,  and  in  various  ways, 
preventable  or  non-preventable. 

The  other  element  in  the  interest-rate,  the  value  of  the  use  of  cap- 
ital, also  fluctuates,  as  in  times  of  capital  famin  or  capital  glut,  in  new 
countries  as  against  old  countries,  and  it  is  difficult  to  decide  how  much 
of  the  rate  is  due  to  this  source  and  how  much  to  insurance.  But  taking 
the  rate  as  a  whole,  the  question  is,  does  the  price  of  a  bond  ever  vary 
except  thru  the  interest-rate  ? 

We  may  test  this  by  experiment.  Taking  some  railroad  company 
which  has  fallen  into  misfortune  and  whose  S}4  per  cent,  bonds,  once  at 
a  premium,  are  now  below  par,  so  that  their  present  market  price  is 
equivalent  to  discounting  all  the  recipiends  at  4.50,  If  this  depreciation 
is  not  entirely  a  consequence  of  this  high  rate  of  discount,  if  there  is  an 
intrinsic  depreciation,  it  must  apply  to  all  obligations  of  the  road.  But 
if  the  same  road  now  puts  out  bonds  bearing  5  per  cent,  interest  under 
the  same  mortgage  it  will  invariably  be  found  that  these  will  sell  at  a 
premium,  on  approximately  the  same  (4>^)  basis. 

We  may  deduce  the  following  conclusions  : 

1.  There  is  no  sanctity  in  par  ;  it  is  merely  a  convenient  round  sum 
to  be  received  in  the  future. 

2.  There  is  no  necessary  identity  between  the  size  of  the  coupons, 
or  periodical  instalments,  and  the  rate  of  interest. 

3.  All  the  recipiends  (coupons  and  par)  must  be  sold  below  par  ; 
their  aggregate  may  amount  to  more  or  less  than  par  or  to  exactly  par. 

4.  The  rate  of  interest  is  affected  by  the  degree  of  belief  in  the 
certainly  and  punctuality  of  the  payments  ;  and  this  rate  determins  the 
price. 

It  may  further  be  stated  that  no  investment  is  so  insecure  that, 
theoretically,  it  will  not  be  discounted  at  some  rate.  A  $1000  bond 
secured  by  something  which  must  be  annihilated  at  the  end  of  five  years, 
but  bearing  30  per  cent,  semi-annual  coupons,  would  doubtless  find  pur- 
chasers at  better  than  $400,  and  would  be  an  advantageous  purchase. 


Amortization  27 

The  market  value  is  of  absolutely  no  importance  to  an  investor 
who  does  not  contemplate  changing  the  investment,  but  will  hold  it  to 
maturity.  The  ups  and  downs  of  the  market  do  not  in  the  least  affect 
the  value  to  him  ;  if  he  were  to  record  these  fluctuations  it  would  be 
merely  to  substitute  an  undulating  zig-zag  for  the  natural  and  logical 
curve  of  the  investment  value,  since  in  either  case  the  point  of  final  rest 
is  par  at  maturity.  Such  a  case  is  that  of  a  trustee  who,  under  the 
decisions  of  the  courts  of  New  York,  is  bound  to  keep  his  trust  intact, 
carrying  the  investments  at  their  investment  value  and  re-investing  all 
in  excess. 

But  to  the  investor  who  has  the  privilege  of  selling  and  replacing 
his  investments,  acquaintance  with  market  values  is  highly  advanta- 
geous. It  is  his  guide  to  the  advisability  of  making  such  changes  and  of 
forecasting  the  future.  It  is  his  duty,  therefore,  to  watch  the  fluctuations 
of  the  market  and,  in  a  perfect  legitimate  way,  seek  to  improve  his  in- 
come, without  impairing  the  factor  of  safety.  A  large  investor  will  not 
endeavor  to  have  all  his  investments  at  the  same  grade  ;  he  will  probably 
have  at  the  same  time  some  capital  out  at  high  rates  and  some  at  lower. 
The  money  at  high  rates  is  not  quite  so  secure,  not  quite  so  available, 
and  requires  more  effort  for  its  collection.  That  at  lower  rates  is  nearer 
to  absolute  freedom  from  risk  and  from  labor  ;  it  almost  automatically 
collects  its  own  income.  On  some  of  the  high-interest  investments  the 
security  may  have  improved  in  the  course  of  time ;  the  credit  of  the 
municipality  or  the  revenue  of  the  corporation  may  have  so  risen  that 
the  4  per  cent,  bond,  which  was  bought  at  par,  is  now  selling  at  a  pre- 
mium which,  if  a  further  investment  were  made,  would  yield  only  S}4 
per  cent.  If  the  bond  has  still  ten  years  to  run,  he  may  sell  at  a  profit  of 
4  per  cent,  and  thus  have  |104  to  re-invest  in  some  other  4  per  cent, 
security. 

Altho  the  market  price  is  of  great  utility,  I  do  not  admit  that  it 
can  be  introduced  into  the  accountancy  of  investment.  It  is  not  an  act 
nor  a  fact  of  the  business  ;  it  is  a  statement  of  what  might  be  done.  When 
the  bond  just  mentioned  has  gone  up  to  104,  the  owner  has  not  gained  a 
penny.  He  merely  has  an  opportunity  presented  ;  if  he  lets  it  pass,  the 
opportunity  has  not  had  the  slightest  effect  on  his  financial  status. 

Unless  the  accounts  are  kept  on  the  investment-value  basis,  he 
cannot  even  tell  whether  a  certain  price  would  result  in  a  loss  or  a  gain. 
If  his  books  are  kept  on  the  basis  of  par,  every  sale  above  par  will  appear 
as  a  gain,  tho  it  may  be  a  losing  bargain  ;  while  a  comparison  with  origi- 
nal coat  will  be  equally  delusiv. 

Where  liquidation,  entire  or  partial,  is  a  possible  contingency,  as 
in  a  savings  bank  or  an  insurance  company,  market  values  are  an  appro- 
priate basis  for  an  estimate  of  solvency.  It  must  be  remembered,  how- 
ever, that  solvency  for  going  on  and  solvency  for  winding  up  are  different 
matters  and  that  in  a  going  concern,  going  insolvency  is  primarily  to  be 
considered. 


28  Amortization 

My  conclusions  as  to  the  proper  basis  of  accountancy  for  an  investor 
are  as  follows  : 

1.  Neither  original  cost  nor  ultimate  par  is  a  proper  permanent 
basis,  but  the  bond  should  enter  at  cost,  which  is  a  fact,  and  should  go 
out  at  par,  which  is  another  fact. 

2.  During  the  interim  the  reduction  from  cost  to  par  should  take 
place  gradually  by  the  processes  of  amortization  and  accumulation  at  the 
basis-rate  of  true  interests 

3.  Information  should  be  obtained  of  the  fluctuations  in  market 
value,  but  these  should  not  be  carried  into  the  accounts  as  actualities. 

4.  A  list  of  market  values  should  accompany  the  balance-sheet  of 
any  concern  which  may  be  subject  to  liquidation  for  the  purpose  of 
showing  its  ability  to  liquidate. 


Address 


t 

wn-H 


E    M 


REMITTAN(  '.e 


ElOOKS 

ATH 


Text  Book  of  the  Accountancy 

"three  parts  i 
la.   The  Accountancy 


<)n  a  large 
yaluation  of 


lb. 


Problems  and 

above,  contai 
fliflicult  cases 

ic.    Tables  of  Compound 

i>i  decimals, 

Complete  ixtended 

Eight  places 
2a.    [Not  isold  separa 
2b.     [Soldj  separately] 

The  Philosophy  of  A(*:ounts 

Amortizatipn 

12-piace  Logfarithms 


ON 
MAT 

CHARLES 


ISi! 


of 

1  one  ;  sold  s 

of  Investme:<it 

sc41e,  a  treati 
tonds,  arithn: 

Stiudies   in 

ingnumeroas 
;  abbreviatec  I 

Interest, 
Uniform  w 

Tables 

)f  decimals. 

2l4^  to; 

lK*t0  2^^ 


Bmd 


tely] 


(In  press) 


ACCOU 


Accountai|cy 

examples 
methods 


>lscoant,  Si^kingr 

ih  two  work 


1 14  <jci  io  10?^  t  asis 


-  si  to  10^ 


NTANCY    AND 

CS     OF     Fl 

E.    SPRAGUE 

54   WEST   32nd 


NANQE 

"ORK.    N.  Y 


Investmeni 

;parately 
:    Comprising,  besides  full  details  of 
on  compou  nd  interest,  {discount 

etical  explajiation  of  allsprocesses 

of  Investment:    a  s 

or  practice  vith  their 


Funds^  Annuities 

>  above 


ST.,    NEW 


Price  f4.oo 

the  book-ketping  of  securities 

annuities,  sinking  funds  anid  the 

Price  $2.oo 

uppleraentai-y  volume  to  the 

solutions  ;  elabcjration  of  spe(  ially 

Price  I1.50 

etc.:    Calculated  to   8  places 

Price  $1.00 

Price  I J  0.00 


Price 
Price 
Price  50 
Price 


l3. 


I3-00 
00 

dents 
00 


13. 


-^T^T-VFT'^TT 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN     INITIAL     FINE     OF     25     CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.00  ON  THE  SEVENTH  DAY 
OVERDUE. 


DEC     2   1932 

'''   S     t932 

*'0«'    6    1934 
APR  30  1938 


MAY    14  19C0 
"•^i    28  192 J 


]a-    fi-b^ 


'/( 


DfC 


^61940M 


LD  21-50m-8,-32 


PAMPHLET  BINDER 

Syracuse,  N.   Y. 
Stockton,  Colif. 


